ACT · Math · Preparing for Higher Math: Algebra

Inequalities

Inequalities is part of Preparing for Higher Math: Algebra in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include solve a linear inequality, compound inequality, inequality from a context and count integer solutions.

Common traps: not reversing the inequality for a negative; reversing the inequality when no negative is used; dropping the equal sign.

Sample questions

Question 1
Solving −(2/3)x + 4 ≥ 10 for x gives which of the following?
- A.
  x ≥ −9
- B.
  x ≤ −4
- C.
  x ≥ 9
- D.
  x ≤ −9
Answer: D. x ≤ −9
Subtract 4: −(2/3)x ≥ 6. Multiply by −3/2 and reverse the sign: x ≤ −9. A does not reverse; B multiplies 6 by −2/3 instead of dividing by it; C ignores the negative coefficient altogether, solving (2/3)x ≥ 6.
Question 2
For which values of x is 2(3 − x) ≥ 4x − 18?
- A.
  x ≤ −2
- B.
  x ≥ 4
- C.
  x ≤ 4.8
- D.
  x ≤ 4
Answer: D. x ≤ 4
6 − 2x ≥ 4x − 18 gives 24 ≥ 6x, so x ≤ 4. B comes from writing −6x ≥ −24 and dividing by −6 without reversing; A moves −18 as −18 instead of +18 (−12 ≥ 6x); C multiplies only the 3 by 2 (6 − x ≥ 4x − 18).
Question 3
A number x satisfies both 2x − 5 < 9 and 3 − x ≤ 1. Which of the following describes all such x?
- A.
  −2 ≤ x < 7
- B.
  2 < x ≤ 7
- C.
  x ≤ 2
- D.
  2 ≤ x < 7
Answer: D. 2 ≤ x < 7
2x − 5 < 9 gives x < 7. 3 − x ≤ 1 gives −x ≤ −2, so x ≥ 2 after reversing. Together, 2 ≤ x < 7. C forgets to reverse the second inequality (x ≤ 2 and x < 7 means x ≤ 2); A drops a sign; B swaps the endpoint inclusion.
Question 4
Aarav's first three science quiz scores are 78, 85 and 90 out of 100. What is the least score he needs on the fourth quiz for the mean of all four scores to be at least 85?
- A.
  2
- B.
  84
- C.
  85
- D.
  87
Answer: D. 87
The four scores must total at least 4 × 85 = 340. He has 78 + 85 + 90 = 253, so he needs at least 340 − 253 = 87. 2 uses 3 × 85 instead of 4 × 85; 84 is his current mean, rounded down; 85 assumes he only needs to score the target mean.
Question 5
What is the greatest integer value of x that satisfies -4x - 7 > 10?
- A.
  -5
- B.
  -4
- C.
  -3
- D.
  5
Answer: A. -5
Add 7: -4x > 17. Dividing by the negative number -4 reverses the inequality: x < -17/4. The greatest integer less than -17/4 is -5.

Find your real gaps

A few questions show the style. The free scan shows which ACT skills cost you the most points, then builds a plan around them.

Take the free scan